Matrix Representations of Linear Map with Respect to Orthonormal Basis

Theorem

1. Let $T\in \mathcal{L}(v)$
2. Suppose $\beta =\{{\alpha }_1,\dots ,{\alpha }_n\}$ is an ordered Orthonormal Basis for $V$
3. Let $A=[T{]}_{\beta }=A_{kj}$ where $1\le k,j\le n$
4. Then, $A_{kj}:=⟨T{\alpha }_i|{\alpha }_{k⟩}$

Example

With $T(x,y)=(x+y,x-y)$ With Dot Product With basis $\{e_1,e_2\}$ Then, $$[T] = \left[\begin{array}{cc} \langle T e_{1} | e_{2} \rangle & \langle T e_{2} | e_{1} \rangle\ \langle T e_{2} | e_{1} \rangle & \langle T e_{2} | e_{2} \rangle\ \end{array}\right] = \left[\begin{array}{cc} \langle (1 , 1) | (1,0) \rangle & \langle (1, -1) | (1,0) \rangle\ \langle (1 , 1) | (0,1) \rangle & \langle (1, -1) | (0,1) \rangle\ \end{array}\right] = \left[\begin{array}{cc} 1 & 1\ 1 & -1\ \end{array}\right]